Cohomological Hall Algebras of Calabi-Yau 3-folds

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"Calabi-Yau 3-folds" are 6-dimensional curved spaces that are important in Geometry, and also in String Theory in Theoretical Physics. String Theory is consistently defined only in 10-dimensional spacetime, and in order to describe our 4-dimensional spacetime (3 space dimensions and 1 time dimension), one is required to wrap the additional 6 dimensions on a very small Calabi-Yau 3-fold. The geometry of the Calabi-Yau 3-fold determines our 4-dimensional physics (particles, etc). By associating a 4-dimensional physical theory to the Calabi-Yau 3-fold, which is not mathematically understood, String Theorists make amazing conjectures about Calabi-Yau 3-folds, an area known as Mirror Symmetry."Donaldson-Thomas invariants" are numbers counting geometric objects (coherent sheaves) living on a Calabi-Yau 3-fold X. The coherent sheaves form a "moduli space" M, a singular space, and DT invariants are defined by an unusual kind of integration over M. In String Theory, DT invariants are "numbers of BPS states", a kind of particle. In 2008, the PI and Yinan Song showed how to define DT invariants in the most general case, and proved they change by a "wall-crossing formula" as the structure on X deforms. This led to an explosion of research on Donaldson-Thomas theory and its extensions. In 2013, the PI showed DT invariants can be interpreted as dimensions of vector spaces. The vector spaces have a difficult construction as a kind of exotic cohomology of the moduli space M. (Cohomology measures the "shape" of a space M, e.g. the hole in a donut.) In String Theory, these vector spaces are "vector spaces of BPS states", part of the Quantum Field Theory associated to X.It is a long standing conjecture in Geometry (Kontsevich-Soibelman) and String Theory (Harvey-Moore) that these vector spaces should have a multiplication on them making them into an algebra (something with addition and multiplication, like ordinary numbers) - the "algebra of BPS states" in the Physics literature, or "Cohomological Hall Algebra (CoHA)" in the mathematics literature. In Physics, the multiplication comes from two particles joining to make a third particle. The conjecture was proved in 2010 by Kontsevich-Soibelman for "quivers with superpotential", a toy model for Calabi-Yau 3-folds. Work by the PI and Pavel Safronov in 2015 enables one to define the multiplication over small regions of the moduli space M, but not yet over the whole space.In this proposal, we aim to construct the multiplication on the vector space of BPS states. We will do this by proving a much more general conjecture of the PI from 2013 in the area of Shifted Symplectic Derived Algebraic Geometry, by a new method.Proving this conjecture enables us to define CoHAs for Calabi-Yau 3-folds, which are infinite-dimensional algebras, and DT invariants are dimensions of pieces of these algebras. We can then study these algebras using Representation Theory, e.g. it may be possible to show that DT invariants are power series coefficients of modular forms, a class of special functions in Number Theory.The conjecture we will prove also has other very important applications, which we explore in the proposal:* It gives an alternative construction of "DT4 invariants" of 8-dimensional Calabi-Yau 4-folds X, defined by Joyce and Borisov in 2015, and shows these DT4 invariants have additional useful properties, e.g. how they behave on cutting X into 2 pieces.* It allows us to define an algebraic geometry version of the "Fukaya category" of a symplectic manifold, which are key to Mirror Symmetry. This algebraic version will be simpler and more rigid than the usual version, and work without many of the usual restrictive assumptions.* This "Fukaya category" has important applications, including to the study of knots in ordinary 3-dimensional space, and to making invariants of knots into a mathematical structure called a Topological Quantum Field Theory.